Graphs of Sine and Cosine Functions

Lecture 28: Section 4.5 Graphs of Sine and Cosine Functions Period of sine and cosine Amplitude of sine and cosine Horizontal translation - phase shift Vertical translations

L28 - 1

THINKABOUTTHE UNIT CIRCLE

The Graph of y = sin x

Domain:

ANY X VALUE YOU WANT

Range: 7,10

Period: 21T STARTS REPEATING THE CYCLE AFTER 21T The key points in one period:

12

oil

x

P

ooo 10

3

0

2

2

2

01

y = sin x 0

312 2

I

O

I

0

1

2

2

2

-1

L28 - 2

The Graph of y = cos x

Domain: C P P ANY X VALUE YOU WANT Range: C I I Period: 21T The key points in one period:

12 0,1

x

o 1110

3

0

2

2

2

oil

y = cos x I

0

I

3I

2

O

l

To1

F

Y

z

2

-1

Checkpoint: Lecture 28, problem 1

L28 - 3

Amplitude for Functions y = a sin x and

y = a cos x

i

Def. The amplitude of y = a sin x and y = a cos x represents half the distance between the maximum and minimum values of the function and is given by

Amplitude = |a|

maximum value - minimum value

=

20

NOTE:

1. If |a| > 1, the curve is stretched vertically, and if |a| < 1, the curve is shrunk vertically.

2. The range is [-a, a].

1 ex. Graph y = sin x

2

AMP _141 1

1

DOMAIN 0,0

ZANGE I I

-

-1

1 g Sink

2 y fsin x

90T EACH

y VALUE IN

HALF

y

2i

i

i

L28 - 4

MULTIPLY EACH Y VA

ex. Graph y = -2 cosBxYZ

y COSCX 7 2 y 2C0S X

3 y 2C X

i

1a l e al

oo

fol

r

r

I rr

-

i

2

-1

X

r

I

l

rr

og

o

AMP

21 2

MAX MIN

I1

OR AMP 2

42 2

DOMAIN P A

RANHE f 2,2

Checkpoint: Lecture 28, problem 2

L28 - 5

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